MATHEMATICAL ANALYSIS OF FRACTIONAL ORDER CO-INFECTION TB AND HIV MODEL

. A mathematical model of HIV/AIDS and TB including its co-infections is formulated. We ﬁnd the Equilibrium points and with the help of numerical simulation, we have analyzed that the sub-models of TB, HIV/AIDS and its co-infections. The Caputo and Caputo Febrizo fractional derivative operator of order α ∈ (0 , 1] is employed to obtain the system of fractional diﬀerential equations. Laplace Adomian Decomposition Method was successfully used for solving the diﬀerent diﬀerential equations.Laplace transform is a perfect technique in various ﬁeld of biological science,engineering,pure and applied mathematics. The latest technique Laplace Adomian Decomposition Method is employed on the developed fractional order model for the numerical solutions. Finally numerical simulations are also established to investigate the inﬂuence of the system parameter on the spread of disease and which show eﬀect of fractional parameter α on our obtained solutions.


Introduction
Mathematics plays a key role in many scientific disciplines, primarily as a mathematical modeling tool.
Mathematics may be thought to play an equally important or even less important role in biology, and mathematics and biology may have something in common. Although physics and biology are very different sciences, mathematicians and biologists have developed "mathematical biology" or "biomathematics" in the latest disciplines of biological mathematical characterization, as well as research in biology, biomedical and biotechnological sciences. The emphasis on biomathematics by the current departmental name is undoubtedly a response to the rapid growth in the use of mathematics in biology in recent years [1], but it is in line with my official duties, including the explicit mention of statistics. The mathematician has taken initiatives, but there is no precedent. As Prof. EJ Williams of Melbourne stated in his review of the development of Biomathematical models [2], a mathematical model of any phenomenon can only contribute to understanding when it comes to conclusions that can be tested at using experimental or experimental evidence. Operational test theory is nothing more than pure mathematics, even though the subject of the discussion is described in biological terms.
The use of mathematics in biology is important because biology becomes more quantitative. This new discipline has applications in the fields of biology, biomedicine and biotechnology. The goal of this new discipline is to mathematically represent and model biological problems using various theories and mathematical techniques. Mathematical biology has dramatic and practical applications in the fields of biology, biomedicine and biotechnology. The long-term interaction between mathematical methods and biology is explained by the problematic approach: how does the brain work? Mathematics plays a role in brain research, from Von Helmholtz's [3] early work, which looks for functions similar to energy, the physical and chemical foundations that describe brain dynamics, and Freud's catharsis [4]. Norbert Wiener's cybernetics to study biological control mechanisms [5], mathematical research on the nervous system [6] and recent use of mathematics to study consciousness [7,8]. Examine the Dynamical transmission and effect of smoking in society [15]. The distinguished features of fractional differential equations are that it outlines memory and transmitted properties of numerous mathematical models. As a fact, that fractional order models are more realistic and practical than the classical integer order models. Fractional order derivative produces greater degree of freedom in these models. Arbitrary order derivatives are powerful tools for the discretion of the dynamical behavior of various biomaterial and systems [16,17].
Tuberculosis and AIDS Tuberculosis and HIV co-infection refer to people with both HIV infection and latent or active TB. When a person has HIV and TB, each disease accelerates the progression of another disease. In addition to the accelerated progression of HIV infection from latent to active TB, tuberculosis has accelerated the progression of HIV infection [9]. HIV infection and TB bacterial infections are completely different infections. If you are infected with HIV, you will not be infected with TB unless you are in contact with someone who is infected with TB. However, if you live in a country where the prevalence of TB is high, it can happen if you do not realize it. Similarly, if you have TB, you will not be infected with HIV unless you engage in activities such as unprotected sex with people already infected with HIV. Tuberculosis also occurs earlier than many other opportunistic infections during HIV infection. Co-infected people are twice as likely to die as HIV infected people without TB. This is true even when considering antiretroviral treatment (ARV) [10].
The World Health Organization (WHO) estimates that between 2005 and 2012, cooperation activities in tuberculosis prevention and treatment, including TB prevention, antiretroviral therapy and HIV testing, were reinstated at 1.3 million. As a result, these cooperative activities have infected humans by reducing tuberculosis and HIV. In recent years, the number of TB-related deaths among people living with HIV has decreased [11]. Overall, the lifetime risk of HIV-negative patients from latent progression to active progression is estimated at about 5-10 percent. For HIV-positive people, the same number represents a 2-year risk [12]. In 2016, approximately 374,000 people with TB and HIV were killed. This is a supplement to the death of only 1.3 million people with TB [13]. It has been reported that people co-infected with HIV and tuberculosis die as a result of HIV infection [14]. The symptoms of active TB and AIDS make you feel sick. The cough lasts more than 2 weeks, coughing mucus or blood, chest pain. You may also have weakness or fatigue, weight loss, lack of appetite, fever or chills, and night sweats.
The link between HIV and TB is opportunistic infection (OI). For people whose immune systems are weakened, infections occur more often or more severely than those with healthy immune systems. Treatment with anti-HIV drugs is called antiretroviral therapy (ART). Antiretroviral therapy protects the immune system and prevents HIV infection from contracting AIDS. In people co-infected with HIV and TB, antiviral therapy may reduce the risk of developing latent tuberculosis tuberculosis. When you cough or sneeze, the transfer causes tuberculosis and the bacteria of tuberculosis spread through the air. If you are pregnant, you are also more likely to develop TB. Age is under 5 years or over 65 years old. Drink or inject drugs. Do not eat well. Once inside the body,TB can be inactive or active. Inactive TB is called latent TB. Active TB is called tuberculosis.

Mathematical Model
Firstly split host people represented with T (t) in the form of four sections: susceptible persons w(t), and Here   Hence, the endemic point is
But, if g / ∈H(c,d) under this condition, the derivative is given follows: , then the above equation assumes the form Its is necessary to have the antiderivative associated with the new fractional derivative.
Definition 02: Let us recall the well known definition of Caputo fractional derivative. Given c > 0, j ∈ H 1 (0, c) and 0 < α < 1, the Caputo fractional derivative of j of order α is given by Fractional calculus and, in particular, Caputo fractional derivative, finds numerous applications in different areas of science.
one obtains the new Caputo-Fabrizio fractional derivative of order 0 < α < 1. That is where M (α) is a normalization constant depending on α. According to the new definition, it is clear that if j is a constant function, then C D α j(t) = 0 as in the usual Caputo derivative. The main difference between old and new definition is that, contrary to the old definition, the new kernel has no singularity for t = s. It is well known that Laplace Transform plays an important role in the study of ordinary differential equations.
In the case of this new fractional definition, it is also known that, for 0 < α < 1 where L{j} denotes the Laplace Transform of function j. So, it is clear that if we work with Caputo-Fabrizio derivative, Laplace Transform will also be a very useful tool.
The series solution is as follows

Fractional Order Model with Caputo Febrizio Sense
The fractional order model of the system (2.1-2.14) by using caputo Febrizio derivative is given as.
The series solution is as follows

Numerical Results and Discussion
The mathematical analysis of SEICR epidemic model with nonlinear system of differential equation has been presented.

Conclusion
We developed a scheme for the solution of Co-Infection of TB and HIV by using Laplace Adomian decomposition method. The well known Co-Infection of TB and HIV model namely Susceptible population, Exposed population, Co-infectious population, Quarantined population, Recovered population and Vaccinated is considered with and without demographic effects. We can solved the detail model of Co-Infection of TB and HIV with fractional order derivative with Caputo sense and Caputo fabrizio sense and we use the Laplace Adomian Decomposition method to solve the model. It is also observed that to eliminate disease, it is not necessary to vaccinate whole of population. The effect of fractional parameter on our obtained solutions is presented through Graphs It is worthy to observe that fractional order derivative show significant changes and memory effect as compared to ordinary derivative.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.